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log(2cosx)=2 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
log(2*cos(x)) = 2
log⁡(2cos⁡(x))=2\log{\left(2 \cos{\left(x \right)} \right)} = 2
Simplificar
Solución detallada
Tenemos la ecuación
log⁡(2cos⁡(x))=2\log{\left(2 \cos{\left(x \right)} \right)} = 2
cambiamos
log⁡(2cos⁡(x))−2=0\log{\left(2 \cos{\left(x \right)} \right)} - 2 = 0
log⁡(2cos⁡(x))−2=0\log{\left(2 \cos{\left(x \right)} \right)} - 2 = 0
Sustituimos
w=cos⁡(x)w = \cos{\left(x \right)}
Tenemos la ecuación
log⁡(2w)−2=0\log{\left(2 w \right)} - 2 = 0
log⁡(2w)=2\log{\left(2 w \right)} = 2
Es la ecuación de la forma:
log(v)=p

Por definición log
v=e^p

entonces
2w=e212 w = e^{\frac{2}{1}}
simplificamos
2w=e22 w = e^{2}
w=e22w = \frac{e^{2}}{2}
hacemos cambio inverso
cos⁡(x)=w\cos{\left(x \right)} = w
Tenemos la ecuación
cos⁡(x)=w\cos{\left(x \right)} = w
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
x=πn+acos⁡(w)x = \pi n + \operatorname{acos}{\left(w \right)}
x=πn+acos⁡(w)−πx = \pi n + \operatorname{acos}{\left(w \right)} - \pi
O
x=πn+acos⁡(w)x = \pi n + \operatorname{acos}{\left(w \right)}
x=πn+acos⁡(w)−πx = \pi n + \operatorname{acos}{\left(w \right)} - \pi
, donde n es cualquier número entero
sustituimos w:
Gráfica
0-80-60-40-2020406080-100100-1010
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Suma y producto de raíces [src] 
suma
           /    / 2\\       /    / 2\\     /    / 2\\
           |    |e ||       |    |e ||     |    |e ||
2*pi - I*im|acos|--|| + I*im|acos|--|| + re|acos|--||
           \    \2 //       \    \2 //     \    \2 //
(2π−iim⁡(acos⁡(e22)))+(re⁡(acos⁡(e22))+iim⁡(acos⁡(e22)))\left(2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{e^{2}}{2} \right)}\right)}\right) + \left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{e^{2}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{e^{2}}{2} \right)}\right)}\right) 
=
         /    / 2\\
         |    |e ||
2*pi + re|acos|--||
         \    \2 //
re⁡(acos⁡(e22))+2π\operatorname{re}{\left(\operatorname{acos}{\left(\frac{e^{2}}{2} \right)}\right)} + 2 \pi 
producto
/           /    / 2\\\ /    /    / 2\\     /    / 2\\\
|           |    |e ||| |    |    |e ||     |    |e |||
|2*pi - I*im|acos|--|||*|I*im|acos|--|| + re|acos|--|||
\           \    \2 /// \    \    \2 //     \    \2 ///
(2π−iim⁡(acos⁡(e22)))(re⁡(acos⁡(e22))+iim⁡(acos⁡(e22)))\left(2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{e^{2}}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{e^{2}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{e^{2}}{2} \right)}\right)}\right) 
=
/           /    / 2\\\ /    /    / 2\\     /    / 2\\\
|           |    |e ||| |    |    |e ||     |    |e |||
|2*pi - I*im|acos|--|||*|I*im|acos|--|| + re|acos|--|||
\           \    \2 /// \    \    \2 //     \    \2 ///
(2π−iim⁡(acos⁡(e22)))(re⁡(acos⁡(e22))+iim⁡(acos⁡(e22)))\left(2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{e^{2}}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{e^{2}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{e^{2}}{2} \right)}\right)}\right) 
(2*pi - i*im(acos(exp(2)/2)))*(i*im(acos(exp(2)/2)) + re(acos(exp(2)/2)))
Respuesta rápida [src] 
                /    / 2\\
                |    |e ||
x1 = 2*pi - I*im|acos|--||
                \    \2 //
x1=2π−iim⁡(acos⁡(e22))x_{1} = 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{e^{2}}{2} \right)}\right)} 
         /    / 2\\     /    / 2\\
         |    |e ||     |    |e ||
x2 = I*im|acos|--|| + re|acos|--||
         \    \2 //     \    \2 //
x2=re⁡(acos⁡(e22))+iim⁡(acos⁡(e22))x_{2} = \operatorname{re}{\left(\operatorname{acos}{\left(\frac{e^{2}}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{e^{2}}{2} \right)}\right)} 
x2 = re(acos(exp(2)/2)) + i*im(acos(exp(2)/2))
Respuesta numérica [src] 
x1 = 25.1327412287183 + 1.98115964675051*i
x2 = -251.327412287183 - 1.98115964675051*i
x3 = 81.6814089933346 + 1.98115964675051*i
x4 = -56.5486677646163 - 1.98115964675051*i
x5 = 31.4159265358979 - 1.98115964675051*i
x6 = 87.9645943005142 - 1.98115964675051*i
x7 = 622.035345410779 + 1.98115964675051*i
x8 = -62.8318530717959 + 1.98115964675051*i
x9 = -81.6814089933346 - 1.98115964675051*i
x10 = 94.2477796076938 - 1.98115964675051*i
x11 = 75.398223686155 - 1.98115964675051*i
x12 = -37.6991118430775 + 1.98115964675051*i
x13 = 56.5486677646163 - 1.98115964675051*i
x14 = 62.8318530717959 + 1.98115964675051*i
x15 = -18.8495559215388 + 1.98115964675051*i
x16 = -100.530964914873 - 1.98115964675051*i
x17 = 6.28318530717959 + 1.98115964675051*i
x18 = -282.743338823081 + 1.98115964675051*i
x19 = 100.530964914873 - 1.98115964675051*i
x20 = -94.2477796076938 + 1.98115964675051*i
x21 = 5.93472984109987e-67 - 1.98115964675051*i
x22 = 87.9645943005142 + 1.98115964675051*i
x23 = -50.2654824574367 - 1.98115964675051*i
x24 = 69.1150383789755 + 1.98115964675051*i
x25 = -364.424747816416 - 1.98115964675051*i
x26 = -6.28318530717959 - 1.98115964675051*i
x27 = 1.98115964675051*i
x28 = -87.9645943005142 - 1.98115964675051*i
x29 = 50.2654824574367 - 1.98115964675051*i
x30 = 69.1150383789755 - 1.98115964675051*i
x31 = 1.84368999347033e-35 + 1.98115964675051*i
x32 = 6.28318530717959 - 1.98115964675051*i
x33 = 9003.80454518835 - 1.98115964675051*i
x34 = -87.9645943005142 + 1.98115964675051*i
x35 = -37.6991118430775 - 1.98115964675051*i
x36 = -69.1150383789755 + 1.98115964675051*i
x37 = -75.398223686155 - 1.98115964675051*i
x38 = -12.5663706143592 - 1.98115964675051*i
x39 = -18.8495559215388 - 1.98115964675051*i
x40 = -43.9822971502571 + 1.98115964675051*i
x41 = 131.946891450771 + 1.98115964675051*i
x42 = -6.28318530717959 + 1.98115964675051*i
x43 = 50.2654824574367 + 1.98115964675051*i
x44 = 18.8495559215388 + 1.98115964675051*i
x45 = -50.2654824574367 + 1.98115964675051*i
x46 = -31.4159265358979 - 1.98115964675051*i
x47 = 43.9822971502571 + 1.98115964675051*i
x48 = 12.5663706143592 - 1.98115964675051*i
x49 = -25.1327412287183 + 1.98115964675051*i
x50 = 25.1327412287183 - 1.98115964675051*i
x51 = 43.9822971502571 - 1.98115964675051*i
x52 = -62.8318530717959 - 1.98115964675051*i
x53 = -81.6814089933346 + 1.98115964675051*i
x54 = 37.6991118430775 + 1.98115964675051*i
x55 = 1.24469990737641e-32 + 1.98115964675051*i
x56 = -43.9822971502571 - 1.98115964675051*i
x57 = 94.2477796076938 + 1.98115964675051*i
x57 = 94.2477796076938 + 1.98115964675051*i
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